# Differential Calculus Word Problem?

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A colony of bacteria doubles in population every 20 minutes starting from an initial population size of y0. Let y(t) denote the population at time t.

1. Express y as an exponential function with base 2

2. Express y as an exponential function with base e

3. What differential equation does y satisfy?

4. At what time has the initial population grown by a factor of 3?

A colony of bacteria doubles in population every 20 minutes starting from an initial population size of y0. Let y(t) denote the population at time t.

1. Express y as an exponential function with base 2

2. Express y as an exponential function with base e

3. What differential equation does y satisfy?

4. At what time has the initial population grown by a factor of 3?

##### 2 Answers

#### Explanation:

The time law describing the growth of such a population has the form

with

In term of exponential function

If we derive

Finally the population's triplication time is given by the equation

# y = y_0 2^(0.05t) # # y = y_0 e^(0.05ln2t) # # dy/dt = (0.05ln2)y # # t ~~ 32 # mins

#### Explanation:

**1. Express y as an exponential function with base 2**

**2. Express y as an exponential function with base e**

Take natural logarithms

**3. What differential equation does y satisfy?**

**4. At what time has the initial population grown by a factor of 3?**

We want